Texture showing errors (edges of texture)?
Cause

I think the texture gets blurred either because it is scaled out or scaled in via automatic mipmapping texture.



Blur means the adjacent texels are considered when calculating the color.

It will include any texel outside the UV area. It even wraps around the around the texture's border (which is the cause of the effect you discover).



Solution

In both cases I suggest you shrink the UV-face that you get some border around the face (in UV-Editor).



Reference

See Avoid Seams in Textured Objects